3.89 \(\int \frac{\text{PolyLog}(2,a x^q)}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{8 a q^2 x^q \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right ),\frac{1}{2} \left (4-\frac{1}{q}\right ),a x^q\right )}{d (1-2 q) \sqrt{d x}}-\frac{2 \text{PolyLog}\left (2,a x^q\right )}{d \sqrt{d x}}+\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}} \]

[Out]

(-8*a*q^2*x^q*Hypergeometric2F1[1, (2 - q^(-1))/2, (4 - q^(-1))/2, a*x^q])/(d*(1 - 2*q)*Sqrt[d*x]) + (4*q*Log[
1 - a*x^q])/(d*Sqrt[d*x]) - (2*PolyLog[2, a*x^q])/(d*Sqrt[d*x])

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Rubi [A]  time = 0.0599145, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac{2 \text{PolyLog}\left (2,a x^q\right )}{d \sqrt{d x}}-\frac{8 a q^2 x^q \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right );\frac{1}{2} \left (4-\frac{1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt{d x}}+\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q]/(d*x)^(3/2),x]

[Out]

(-8*a*q^2*x^q*Hypergeometric2F1[1, (2 - q^(-1))/2, (4 - q^(-1))/2, a*x^q])/(d*(1 - 2*q)*Sqrt[d*x]) + (4*q*Log[
1 - a*x^q])/(d*Sqrt[d*x]) - (2*PolyLog[2, a*x^q])/(d*Sqrt[d*x])

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{\text{Li}_2\left (a x^q\right )}{(d x)^{3/2}} \, dx &=-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}-(2 q) \int \frac{\log \left (1-a x^q\right )}{(d x)^{3/2}} \, dx\\ &=\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}+\frac{\left (4 a q^2\right ) \int \frac{x^{-1+q}}{\sqrt{d x} \left (1-a x^q\right )} \, dx}{d}\\ &=\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}+\frac{\left (4 a q^2 \sqrt{x}\right ) \int \frac{x^{-\frac{3}{2}+q}}{1-a x^q} \, dx}{d \sqrt{d x}}\\ &=-\frac{8 a q^2 x^q \, _2F_1\left (1,\frac{1}{2} \left (2-\frac{1}{q}\right );\frac{1}{2} \left (4-\frac{1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt{d x}}+\frac{4 q \log \left (1-a x^q\right )}{d \sqrt{d x}}-\frac{2 \text{Li}_2\left (a x^q\right )}{d \sqrt{d x}}\\ \end{align*}

Mathematica [C]  time = 0.0290786, size = 48, normalized size = 0.49 \[ -\frac{x G_{4,4}^{1,4}\left (-a x^q|\begin{array}{c} 1,1,1,1+\frac{1}{2 q} \\ 1,0,0,\frac{1}{2 q} \\\end{array}\right )}{q (d x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[PolyLog[2, a*x^q]/(d*x)^(3/2),x]

[Out]

-((x*MeijerG[{{1, 1, 1, 1 + 1/(2*q)}, {}}, {{1}, {0, 0, 1/(2*q)}}, -(a*x^q)])/(q*(d*x)^(3/2)))

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Maple [C]  time = 0.224, size = 121, normalized size = 1.3 \begin{align*} -{\frac{1}{q}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{1}{2\,q}}} \left ( -4\,{\frac{{q}^{2}\ln \left ( 1-a{x}^{q} \right ) }{\sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}-2\,{\frac{q \left ( 1-2\,q \right ){\it polylog} \left ( 2,a{x}^{q} \right ) }{ \left ( 2\,q-1 \right ) \sqrt{x}} \left ( -a \right ) ^{-1/2\,{q}^{-1}}}-4\,{q}^{2}{x}^{q-1/2}a \left ( -a \right ) ^{-1/2\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,1/2\,{\frac{2\,q-1}{q}} \right ) \right ) \left ( dx \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^q)/(d*x)^(3/2),x)

[Out]

-1/(d*x)^(3/2)*x^(3/2)*(-a)^(1/2/q)/q*(-4*q^2/x^(1/2)*(-a)^(-1/2/q)*ln(1-a*x^q)-2*q/(2*q-1)/x^(1/2)*(-a)^(-1/2
/q)*(1-2*q)*polylog(2,a*x^q)-4*q^2*x^(q-1/2)*a*(-a)^(-1/2/q)*LerchPhi(a*x^q,1,1/2*(2*q-1)/q))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 8 \, q^{3} \int \frac{1}{{\left (2 \, d^{\frac{3}{2}} q +{\left (2 \, a^{2} d^{\frac{3}{2}} q + a^{2} d^{\frac{3}{2}}\right )} x^{2 \, q} - 2 \,{\left (2 \, a d^{\frac{3}{2}} q + a d^{\frac{3}{2}}\right )} x^{q} + d^{\frac{3}{2}}\right )} x^{\frac{3}{2}}}\,{d x} + \frac{2 \,{\left (\frac{{\left ({\left (2 \, a \sqrt{d} q + a \sqrt{d}\right )} x x^{q} -{\left (2 \, \sqrt{d} q + \sqrt{d}\right )} x\right )}{\rm Li}_2\left (a x^{q}\right )}{x^{\frac{3}{2}}} - \frac{2 \,{\left ({\left (2 \, a \sqrt{d} q^{2} + a \sqrt{d} q\right )} x x^{q} -{\left (2 \, \sqrt{d} q^{2} + \sqrt{d} q\right )} x\right )} \log \left (-a x^{q} + 1\right )}{x^{\frac{3}{2}}} + \frac{4 \,{\left (2 \, \sqrt{d} q^{3} x -{\left (2 \, a \sqrt{d} q^{3} + a \sqrt{d} q^{2}\right )} x x^{q}\right )}}{x^{\frac{3}{2}}}\right )}}{2 \, d^{2} q + d^{2} -{\left (2 \, a d^{2} q + a d^{2}\right )} x^{q}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q)/(d*x)^(3/2),x, algorithm="maxima")

[Out]

8*q^3*integrate(1/((2*d^(3/2)*q + (2*a^2*d^(3/2)*q + a^2*d^(3/2))*x^(2*q) - 2*(2*a*d^(3/2)*q + a*d^(3/2))*x^q
+ d^(3/2))*x^(3/2)), x) + 2*(((2*a*sqrt(d)*q + a*sqrt(d))*x*x^q - (2*sqrt(d)*q + sqrt(d))*x)*dilog(a*x^q)/x^(3
/2) - 2*((2*a*sqrt(d)*q^2 + a*sqrt(d)*q)*x*x^q - (2*sqrt(d)*q^2 + sqrt(d)*q)*x)*log(-a*x^q + 1)/x^(3/2) + 4*(2
*sqrt(d)*q^3*x - (2*a*sqrt(d)*q^3 + a*sqrt(d)*q^2)*x*x^q)/x^(3/2))/(2*d^2*q + d^2 - (2*a*d^2*q + a*d^2)*x^q)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}{\rm Li}_2\left (a x^{q}\right )}{d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)*dilog(a*x^q)/(d^2*x^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x**q)/(d*x)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (a x^{q}\right )}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q)/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate(dilog(a*x^q)/(d*x)^(3/2), x)